Theorem 0nnq 10340

Description: The empty set is not a positive fraction. (Contributed by NM, 24-Aug-1995.) (Revised by Mario Carneiro, 27-Apr-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
0nnq	⊢ ¬ ∅ ∈ Q

Proof of Theorem 0nnq

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0nelxp 5583	. 2 ⊢ ¬ ∅ ∈ (N × N)
2		df-nq 10328	. . . 4 ⊢ Q = {𝑦 ∈ (N × N) ∣ ∀𝑥 ∈ (N × N)(𝑦 ~Q 𝑥 → ¬ (2nd ‘𝑥) <N (2nd ‘𝑦))}
3	2	ssrab3 4056	. . 3 ⊢ Q ⊆ (N × N)
4	3	sseli 3962	. 2 ⊢ (∅ ∈ Q → ∅ ∈ (N × N))
5	1, 4	mto 199	1 ⊢ ¬ ∅ ∈ Q

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2110  ∀wral 3138  ∅c0 4290   class class class wbr 5058   × cxp 5547  ‘cfv 6349  2^nd c2nd 7682  Ncnpi 10260   <_N clti 10263   ~_Q ceq 10267  Qcnq 10268

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-opab 5121  df-xp 5555  df-nq 10328

This theorem is referenced by:  adderpq  10372  mulerpq  10373  addassnq  10374  mulassnq  10375  distrnq  10377  recmulnq  10380  recclnq  10382  ltanq  10387  ltmnq  10388  ltexnq  10391  nsmallnq  10393  ltbtwnnq  10394  ltrnq  10395  prlem934  10449  ltaddpr  10450  ltexprlem2  10453  ltexprlem3  10454  ltexprlem4  10455  ltexprlem6  10457  ltexprlem7  10458  prlem936  10463  reclem2pr  10464